Reduced Biquaternion Windowed Linear Canonical Transform: Properties and Applications

Hehe Yang, Qiang Feng (), Xiaoxia Wang, Didar Urynbassarova and Aajaz A. Teali
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Hehe Yang: School of Mathematics and Computer Science, Yanan University, Yanan 716000, China
Qiang Feng: School of Mathematics and Computer Science, Yanan University, Yanan 716000, China
Xiaoxia Wang: School of Mathematics and Computer Science, Yanan University, Yanan 716000, China
Didar Urynbassarova: National Engineering Academy of the Republic of Kazakhstan, Almaty 050000, Kazakhstan
Aajaz A. Teali: Department of Mathematics, School of Chemical Engineering and Physical Sciences, Lovely Professional University, Jalandhar 144411, Punjab, India

Mathematics, 2024, vol. 12, issue 5, 1-21

Abstract: The quaternion windowed linear canonical transform is a tool for processing multidimensional data and enhancing the quality and efficiency of signal and image processing; however, it has disadvantages due to the noncommutativity of quaternion multiplication. In contrast, reduced biquaternions, as a special case of four-dimensional algebra, possess unique advantages in computation because they satisfy the multiplicative exchange rule. This paper proposes the reduced biquaternion windowed linear canonical transform (RBWLCT) by combining the reduced biquaternion signal and the windowed linear canonical transform that has computational efficiency thanks to the commutative property. Firstly, we introduce the concept of a RBWLCT, which can extract the time local features of an image and has the advantages of both time-frequency analysis and feature extraction; moreover, we also provide some fundamental properties. Secondly, we propose convolution and correlation operations for RBWLCT along with their corresponding generalized convolution, correlation, and product theorems. Thirdly, we present a fast algorithm for RBWLCT and analyze its computational complexity based on two dimensional Fourier transform (2D FTs). Finally, simulations and examples are provided to demonstrate that the proposed transform effectively captures the local RBWLCT-frequency components with enhanced degrees of freedom and exhibits significant concentrations.

Keywords: reduced biquaternions; windowed linear canonical transform; convolution; correlation; fast algorithm (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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